The radiation emitted by a star `A` is `1000` times that of the sun. If the surface temperature of the sun and star `A` are `6000 K` and `2000 K` respectively. The ratio of the radii of the star `A` and the sun is:

Options

A. `300 : 1`
B. `600 : 1`
C. `900 : 1`
D. `1200 : 1`

Explanation

To solve the problem, we will use the Stefan-Boltzmann law, which states that the total energy radiated per unit surface area of a black body is proportional to the fourth power of its absolute temperature.

### Step-by-Step Solution:

1. **Understand the Given Information:**
- The radiation emitted by star A is 1000 times that of the Sun.
- The surface temperature of the Sun, \( T_{Sun} = 6000 \, K \).
- The surface temperature of star A, \( T_{A} = 2000 \, K \).

2. **Apply the Stefan-Boltzmann Law:**
The total power radiated by a star can be expressed as:
\[ E = \sigma A T^4 \]
where \( \sigma \) is the Stefan-Boltzmann constant, \( A \) is the surface area, and \( T \) is the temperature.

3. **Express the Surface Areas:**
The surface area \( A \) of a sphere is given by \( A = 4\pi r^2 \). Therefore, we can write the radiated energy for both the Sun and star A:
\[ E_{Sun} = \sigma (4\pi r_{Sun}^2) T_{Sun}^4 \]
\[ E_{A} = \sigma (4\pi r_{A}^2) T_{A}^4 \]

4. **Set Up the Ratio of Radiated Energies:**
Given that \( E_{A} = 1000 \times E_{Sun} \), we can set up the equation:
\[ \sigma (4\pi r_{A}^2) T_{A}^4 = 1000 \times \sigma (4\pi r_{Sun}^2) T_{Sun}^4 \]

5. **Cancel Out Common Factors:**
The \( \sigma \) and \( 4\pi \) terms cancel out:
\[ r_{A}^2 T_{A}^4 = 1000 \times r_{Sun}^2 T_{Sun}^4 \]

6. **Substitute the Temperatures:**
Substitute \( T_{A} = 2000 \, K \) and \( T_{Sun} = 6000 \, K \):
\[ r_{A}^2 (2000)^4 = 1000 \times r_{Sun}^2 (6000)^4 \]

7. **Calculate the Powers:**
Calculate \( (2000)^4 \) and \( (6000)^4 \):
\[ (2000)^4 = 16 \times 10^{12} \quad \text{and} \quad (6000)^4 = 1296 \times 10^{12} \]

8. **Substitute and Rearrange:**
Substitute these values into the equation:
\[ r_{A}^2 \times 16 \times 10^{12} = 1000 \times r_{Sun}^2 \times 1296 \times 10^{12} \]
Simplifying gives:
\[ r_{A}^2 = 1000 \times r_{Sun}^2 \times \frac{1296}{16} \]

9. **Calculate the Ratio:**
\[ r_{A}^2 = 1000 \times r_{Sun}^2 \times 81 \]
Therefore,
\[ \frac{r_{A}}{r_{Sun}} = \sqrt{1000 \times 81} = \sqrt{81000} = 900 \]

10. **Final Result:**
The ratio of the radii of star A to the Sun is:
\[ \frac{r_{A}}{r_{Sun}} = 30 \]

### Conclusion:
The ratio of the radii of star A to the Sun is \( 30:1 \).